Problem 26 · 2015 Math Kangaroo
Stretch
Number Theory
pythagorean-triplefactor-pairs
How many different triangles ABC whose side lengths are whole numbers are there, if ∠ABC = 90° and AB = 20? (Hint: two triangles are called different if they are not congruent.)
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Answer: D — 4
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Hint 1 of 2
With the right angle at B, AB and BC are the legs; you need integer BC and hypotenuse AC.
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Hint 2 of 2
Use AC² − BC² = 400 and factor it as a difference of squares.
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Approach: solve AC² − BC² = 20² by factoring
- The right angle is at B, so 20² + BC² = AC², i.e. (AC−BC)(AC+BC) = 400.
- Both factors must be even; the pairs (2,200),(4,100),(8,50),(10,40) give positive integer legs (the pair (20,20) gives a zero leg).
- That is 4 non-congruent triangles (D).
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